Write the following in the simplest form: ^ -1 x+ 1+x^2 ,x - Vidyadip

Question

Write the following in the simplest form:
$\tan^{-1}\Big\{\text{x}+\sqrt{1+\text{x}^2}\Big\},\text{x}\in\text{R}$

✓

Answer

Let $\text{x}=\cot\theta$
Now,
$\tan^{-1}\Big\{\text{x}+\sqrt{1+\text{x}^2}\Big\}$
$=\tan^{-1}\Big\{\cot\theta+\sqrt{1+\cot^2\theta}\Big\}$
$=\tan^{-1}\{\cot\theta+\text{cosec}\theta\}$
$=\tan^{-1}\Big\{\frac{\cos\theta+1}{\sin\theta}\Big\}$
$=\tan^{-1}\Bigg\{\frac{2\cos^2\frac{\theta}{2}}{2\sin\frac{\theta}{2}\cos\frac{\theta}{2}}\Bigg\}$
$=\tan^{-1}\Big\{\cot\frac{\theta}{2}\Big\}$
$=\tan^{-1}\Big\{\tan\Big(\frac{\pi}{2}-\frac{\theta}{2}\Big)\Big\}$
$=\Big(\frac{\pi}{2}-\frac{\theta}{2}\Big)$
$=\frac{\pi}{2}-\frac{\cot^{-1}\text{x}}{2}$

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